A fancy way to obtain the binary digits of $759250125\sqrt{2}$

The rational part of a periodic continued fraction

Let $x$ be a periodic continued fraction with the initial block $0$ and the repeating block $c_1,\ldots,c_n$. So $x$ is a quadratic irrational of the form $x=a+\sqrt b$, where $a$, $b$ are rational numbers, $b>0$, $b$ not a square. The numbers $a$ and $\sqrt b$ are uniquely determined by $x$. In general it is difficult to say what the influence of a certain digit of the repeating block on the appearance of $x$ is. We highlight a noteworthy exception from this rule. Indeed, the magnitude of $2a$ is essentially determined by the last digit $c_n$ of the repeating block, the fractional part of $2a…

Infinite products involving binary digit sums

Let $(u_n)_{n\ge 0}$ denote the Thue-Morse sequence with values $\pm 1$. The Woods-Robbins identity below and several of its generalisations are well-known in the literature \begin{equation*}\label{WR}\prod_{n=0}^\infty\left(\frac{2n+1}{2n+2}\right)^{u_n}=\frac{1}{\sqrt 2}.\end{equation*} No other such product involving a rational function in $n$ and the sequence $u_n$ seems to be known in closed form. To understand these products in detail we study the function \begin{equation*}f(b,c)=\prod_{n=1}^\infty\left(\frac{n+b}{n+c}\right)^{u_n}.\end{equation*} We prove some analytical properties of $…

Matching of orbits of certain $N$-expansions with a finite set of digits

In this paper we consider a class of continued fraction expansions: the so-called $N$-expansions with a finite digit set, where $N\geq 2$ is an integer. These \emph{$N$-expansions with a finite digit set} were introduced in [KL,L], and further studied in [dJKN,S]. For $N$ fixed they are steered by a parameter $α\in (0,\sqrt{N}-1]$. In [KL], for $N=2$ an explicit interval $[A,B]$ was determined, such that for all $α\in [A,B]$ the entropy $h(T_α)$ of the underlying Gauss-map $T_α$ is equal. In this paper we show that for all $N\in \mathbb N$, $N\geq 2$, such plateaux exist. In order to show that…

关于 $ 1/m $ 的 $ n $ 进制展开的某些对称性：类数联系

On some symmetries of the base $ n $ expansion of $ 1/m $ : The Class Number connection

本文研究了 $1/m$ 的 $n$ 进制展开的数字对称性，建立了其与虚二次域 $\Q(\sqrt{-nm})$ 类数的关系，并在 $m\equiv 3\mod 4$ 时推导出涉及 $\Q(\sqrt{-m})$ 类数的同余关系。

On a product of certain primes

We study the properties of the product, which runs over the primes, $$\mathfrak{p}_n = \prod_{s_p(n) \, \geq \, p} p \quad (n \geq 1),$$ where $s_p(n)$ denotes the sum of the base-$p$ digits of $n$. One important property is the fact that $\mathfrak{p}_n$ equals the denominator of the Bernoulli polynomial $B_n(x) - B_n$, where we provide a short $p$-adic proof. Moreover, we consider the decomposition $\mathfrak{p}_n = \mathfrak{p}_n^- \cdot \mathfrak{p}_n^+$, where $\mathfrak{p}_n^+$ contains only those primes $p > \sqrt{n}$. Let $ω( \cdot )$ denote the number of prime divisors. We show that $…

基3的Delange函数的局部极值与无穷导数

Local Extrema and Infinite Derivatives of the Delange Function of Base 3

本文研究基3的Delange函数T的局部结构，明确其局部极值点的特征，刻画T具有无穷导数的点的条件，推导有限立方体上边数最小值的性质、T的变差极限及傅里叶系数。

固定递推序列正规向量集合的Borel复杂度

Borel Complexity of the set of vectors normal for a fixed recurrence sequence

本文研究固定递推序列正规向量集合的Borel复杂度，在特定假设下证明其为$\boldsymbolΠ_3^0$-完全，并得出基$α$正规数集合在$|α|$为Pisot数时也是$\boldsymbolΠ_3^0$-完全的新结果。

三个Brillhart-Lehmer-Selfridge素性证明用于Wagstaff数

Three Brillhart-Lehmer-Selfridge primality proofs for Wagstaff numbers

本文基于Brillhart-Lehmer-Selfridge (BLS) N-1准则，对Wagstaff数W_2617、W_10501和W_12391进行了严格验证的素性证明，不依赖ECPP，仅使用经典N-1方法，并利用Cunningham项目表和FactorDB进行因子分解。

Arithmetics in numeration systems with negative quadratic base

We consider positional numeration system with negative base $-β$, as introduced by Ito and Sadahiro. In particular, we focus on arithmetical properties of such systems when $β$ is a quadratic Pisot number. We study a class of roots $β>1$ of polynomials $x^2-mx-n$, $m\geq n\geq 1$, and show that in this case the set ${\rm Fin}(-β)$ of finite $(-β)$-expansions is closed under addition, although it is not closed under subtraction. A particular example is $β=τ=\frac12(1+\sqrt5)$, the golden ratio. For such $β$, we determine the exact bound on the number of fractional digits appearing in arithmetic…

Deterministic elliptic curve primality proving for a special sequence of numbers

We give a deterministic algorithm that very quickly proves the primality or compositeness of the integers N in a certain sequence, using an elliptic curve E/Q with complex multiplication by the ring of integers of Q(sqrt(-7)). The algorithm uses O(log N) arithmetic operations in the ring Z/NZ, implying a bit complexity that is quasi-quadratic in log N. Notably, neither of the classical "N-1" or "N+1" primality tests apply to the integers in our sequence. We discuss how this algorithm may be applied, in combination with sieving techniques, to efficiently search for very large primes. This has a…